Following the discussion at meta.MO, I'm going to post a good answer from the comments (made by JT) as a "community wiki" answer. I should mention that the Rogawski article mentioned by Tommaso says almost nothing about the proof of Ramanujan's conjecture, but it seems to be a very nice introduction to Jacquet-Langlands.
Deligne reduced Ramanujan's conjecture about the growth of tau to the Weil conjectures (in particular, the Riemann hypothesis) applied to a Kuga-Sato variety, in his paper Formes modulaires et representations l-adiques, Seminaire Bourbaki 355. I believe Jay Pottharst has made an English translation available.
Deligne then proved the Weil conjectures in his paper La conjecture de Weil. I.
As far as I know, all known proofs of this conjecture involve the use of cohomology of varieties over finite fields in an essential way.
Added by Emerton: One point to make is that the Weil conjectures (in their basic form, saying that the eigenvalues of Frobenius on the $i$th etale cohomology of a variety over $\mathbb F_q$ have absolute value $q^{i/2}$) apply only to smooth proper varieties. On the other hand, the Kuga-Sato variety is the symmeteric power of the universal elliptic curve over a modular curve, which is not projective. Thus one has to pass to a smooth compactification in order to apply the Weil conjectures, and then hope that this does not mess anything up in the rest of the argument. A certain amount of Deligne's effort in his Bourbaki seminar is devoted to dealing with this issue.
